[Capt Spaulding]

The explanation for the rapid click/weak battery phenomenon is pretty simple. When you turn the key to the start position the battery has enough power to energize the solenoid. Once the contacts in the solenoid close (click #1), the increased demand by the starter causes enough of a voltage drop that the battery can no longer offer enough power to keep the solenoid energized so the solenoid opens/releases (click #2). Once it releases, the battery is no longer being hammered by the starter and once again has enough power to energize the solenoid (click #3).

Rinse/repeat 20 times per second or so. As most have noted, it's a pretty safe bet a new battery will fix it. I would suggest trying to figure out why this one went dead.

edit: The same logic works for the push button starters. The difference is it's the ECU activating the solenoid.